The results of theoretical modeling in Part I are utilized to develop practical recommendations for developing the algorithms for hail detection and determination of its size as well as attenuation correction and rainfall estimation in the presence of hail. A new algorithm for discrimination between small hail (with maximal size of less than 2.5 cm), large hail (with diameters between 2.5 and 5.0 cm), and giant hail with size exceeding 5.0 cm is proposed and implemented for applications with the S-band dual-polarization Weather Surveillance Radar-1988 Doppler (WSR-88D) systems. The fuzzy-logic algorithm is based on the combined use of radar reflectivity Z, differential reflectivity ZDR, and cross-correlation coefficient ?hv. The parameters of the membership functions depend on the height of the radar resolution volume with respect to the freezing level, exploiting the size-dependent melting characteristics of hailstones. The attenuation effects in melting hail are quantified in this study, and a novel technique for polarimetric attenuation correction in the presence of hail is suggested. The use of a rainfallestimator that is based on specific differential phase KDP is justified on the basis of the results of theoretical simulations and comparison of actual radar retrievals at S band with gauge measurements for storms containing large hail with diameters exceeding 2.5 cm.
The first part of this series (Ryzhkov et al. 2013, hereinafter Part I) provides the results of theoretical modeling of polarimetric radar characteristics of melting hail using a one-dimensional thermodynamic model of Rasmussen and Heymsfield (1987) that was generalized for arbitrary initial size distributions of ice particles at the freezing level, where melting starts. Such a model realistically reproduces vertical profiles of various radar variables in hail-bearing storms and their dependences on radar wavelength and maximal hail size. A more sophisticated 2D cloud model of The
Identification of hail is an inherent part of a number of polarimetric classification algorithms suggested for research studies and operational utilization (e.g., Zrni^c and Ryzhkov 1999; Vivekanandan et al. 1999; Lim et al. 2005; Heinselman and Ryzhkov 2006; Marzano et al. 2008; Park et al. 2009;
In this study, a new polarimetric radar algorithm for detection of hail and determination of its size is de- scribed. The algorithm is devised for applications on S-band radars and aims at discrimination between three categories of hail size as required by the NWS. This al- gorithm was developed at the
Hail produces significant attenuation of radar signal that is not well quantified (Battan 1971; Ryzhkov et al. 2007; Tabary et al. 2009; Borowska et al. 2011; Kaltenboeck and Ryzhkov 2013). No reliable methods for attenuation correction of Z or ZDR in hail exist at the moment, as opposed to rain, where the use of differen- tial phase FDP is very efficient (Bringi et al. 1990, 2001). In rain, attenuation-induced biases of Z and ZDR are linearly proportional to FDP with coefficients of pro- portionality a and b, respectively. The factors a and b are generally not constant and are affected by the vari- ability of drop size distributions. There are two ap- proaches to address the variability of a and b. One of them was suggested by Bringi et al. (2001) according to which the optimal values of a and b are sought while assuming that they are constant along the whole radar ray. Another method by Gu et al. (2011) identifies ''hot spots'' along the propagation path (such as strong con- vective cells possibly containing hail) and determines the optimal values of a and b within hot spots, which are treated separately from the rest of the ray. In section 3, the range of variability of the factors a and b in melting hail is established on the basis of the results of the the- oretical modeling in Part I as well as polarimetric radar observations at S and C band. In addition, a new method for attenuation correction in hail is suggested and tested at S band. The method is a further extension of the hot- spot concept described by Gu et al. (2011), which is particularly efficient for quantification of a and b within hail cells.
Although there is strong observational evidence that rainfall estimation algorithms utilizing specific differ- ential phase KDP are preferable over algorithms using Z and/or Z DR for hail-bearing storms (Balakrishnan and Zrni^c 1990a; Aydin et al. 1995; Hubbert et al. 1998; Ryzhkov et al. 2005; Matrosov et al. 2013), accurate estimation of rain in the presence of hail remains a challenge. Numerous observations with operational S-band WSR-88D systems indicate that KDP frequently exceeds 68-78km21 in the storm cores containing large/ giant hail, which produce unrealistically high rain rates estimated from KDP. In section 4, the KDP-based ap- proach for quantification of rain in the presence of hail is further investigated by comparing simulated vertical profiles of rain rate in melting hail and their estimates using the R(KDP) relations at S, C, and X bands. In ad- dition, the comparison between rain gauge accumula- tions and their estimates using the conventional R(Z) and polarimetric R(KDP) algorithms is performed for a storm that produced large hail ( . 2.5 cm) and was observed by the polarimetric WSR-88D near
We emphasize that Part II (this paper) outlines the guiding principles and method for addressing the three important practical issues involving hail-1) determi- nation of its size, 2) attenuation correction, and 3) rainfall estimation in its presence-that are driven by theoretical simulations in Part I and documented observations. Its refinement and validation is left for further studies. For example, a large-scale validation study using the Se- vere Hazards Analysis and Verification Experiment (SHAVE) method (Ortega et al. 2009, 2012) for the hail size discrimination algorithm (HSDA) using the whole polarimetric WSR-88D network will be described in Part III (A. Ryzhkov et al. 2013, unpublished manu- script). The National Mosaic and Multi-Sensor Quanti- tative Precipitation Estimation (NMQ) system (Zhang et al. 2011) will be used as a platform for validating rainfall estimation and attenuation correction in hail-bearing storms on the WSR-88D network.
2. Detection of hail and determination of its size
It is shown in Part I that the relation between Z and ZDR strongly depends on the degree of melting and on the radar wavelength; hence, the methods for hail de- tection should be wavelength-specific and must take into account the height of the radar resolution volume with respect to the melting level. It is also evident that utilizing a sole ''hail differential reflectivity'' parameter HDR (e.g., Depue et al. 2007) may not be sufficient for discrimination between small (,2.5 cm) and large (.2.5 cm) hail (Picca and Ryzhkov 2012). It is in- strumental to use the cross-correlation coefficient rhv along with Z and ZDR (Balakrishnan and Zrni^c 1990b). It can be shown that the models 1 and 2 utilized in Part I correctly reproduce the decrease of rhv in melting hail as well as its wavelength dependence, but the magnitudes of such decreases are generally smaller than those ex- perimentally observed (e.g., Ryzhkov et al. 2011). This result is likely due to the fact that secondary effects (such as roughness of hailstones or abrupt changes in the shape of the particles during collisions) that lead to possible reductions of rhv are not accounted for in the models (Mirkovic et al. 2013). Nonuniform beamfilling resulting from strong vertical gradients of radar vari- ables in severe convective storms is another possible reason for additional reduction of rhv (Ryzhkov 2007). Picca and Ryzhkov (2012) showed that the depression of rhv above the melting layer in the area of major hail growth between 2108 and 2208C may indicate the pres- ence of giant hail with sizes exceeding 5 cm that usually grows in the wet regime. Because such large hailstones reach the ground with only a relatively small decrease in their initial size, the rhv signature aloft may serve as an indicator of giant hail near the surface.
Theoretical simulations in Part I show that Z and ZDR of melting hail are very sensitive to the slope parameter Lh of the initial size distribution of hail aloft. Because Lh is strongly correlated with maximal hail size
In this study, we suggest a fuzzy-logic scheme to dis- tinguish between three categories of hail size: small hail (diameter D , 2.5 cm), large hail (2.5 , D , 5.0 cm), and giant hail (D . 5.0 cm) on the basis of Z, ZDR, and rhv, with the parameters of the membership functions depending on the relative height of the center of the radar resolution volume with respect to the level of zero wet-bulb temperature, where melting starts. At the moment, HSDA is proposed for utilization at S band only, although a similar type of the algorithm can be developed for C band (and eventually for X band) by taking into consideration higher values of ZDR and lower values of rhv for the same size categories of hail because of the more pronounced effects of resonance scattering (Part I; Kaltenboeck and Ryzhkov 2013; Al- Sakka et al. 2013). The choice of the 2.5- and 5.0-cm diameters of hailstones to delineate the three classes of hail is dictated by the requirement of the NWS to discriminate hail that is smaller and larger than 2.5 cm (about 1 in., which is considered to be ''severe'') and by apparent changes in polarimetric signatures following the transition between large and giant hail such as a significant drop in r hv aloft and the appearance of slightly negative ZDR associated with S-band resonance at about 5 cm in dry hail according to Part I (their Figs. 8c,d). In addition, hailstones with diameters in excess of about 5 cm (2 in.) are considered to be ''significantly severe'' by the NWS.
The parameters of the membership functions of HSDA are determined on the basis of the output of the model studies in Ryzhkov et al. (2009) and Part I as well as radar observations. The results of two recent obser- vational studies in central
The classification algorithm uses six sets of member- ship functions corresponding to six height intervals with respect to the melting level (i.e., where the wet-bulb temperature Tw is equal to zero and melting of hail commences). A trapezoidal shape of the membership functions is selected that is similar to the existing oper- ational Next Generation Weather Radar (NEXRAD) hydrometeor classification algorithm (HCA; Park et al. 2009). The stratification of height intervals implies knowledge of the vertical profile of Tw, which can be obtained either from observed soundings or from the output of numerical weather prediction models such as the Rapid Update Cycle (RUC) or High-Resolution Rapid Refresh RUC (HRRR), similar to the winter classification algorithm described in Schuur et al. (2012). Separate membership functions are utilized in the six height intervals:
1) H . H(Tw 52258C),
2) H(Tw 5 08C) , H , H(Tw 52258C),
3) H(Tw 5 08C) 2 1km, H , H(Tw 5 08C),
5) H(Tw 5 08C) 2 3km, H , H(Tw 5 08C) 2 2 km,
6) H , H(Tw 5 08C) 23 km.
The membership functions for these intervals are de- picted in Fig. 1. At H . H(Tw 52258C) (interval 1), Z is the major discriminator between the three cate- gories of hail because ZDR and rhv do not vary much with hail size at higher altitudes. Therefore, the mem- bership functions of ZDR and rhv are the same for all three hail size categories (Fig. 1a). In the second altitude range, between the 08 and 2258C wet-bulb temperature isotherms, Z and rhv have major discriminative power, whereas ZDR weakly depends on maximal diameter of hailstones (Fig. 1b). All three radar variables possess strong classification capability below the freezing level with the discriminative power of ZDR increasing toward the ground (Figs. 1c-f).
The algorithm for hail detection and discrimination of its size is conceived as a natural extension of the existing NEXRAD HCA, which identifies the class ''hail mixed with rain'' (Park et al. 2009). The suggested algorithm splits this class designation into three categories of hail size using the aforementioned fuzzy-logic routine in locations where the HCA recognizes hail mixed with rain. For the time being, the values of all three mem- bership functions are summed up with equal weights, although this can be changed in the future. Similar to the existing WSR-88D HCA, the HSDA algorithm provides classification results at all antenna elevations, and class designation at any particular elevation is performed in- dependently; that is, the algorithm does not utilize full vertical profiles of radar variables. An experimental version of the S-band HSDA has been run on a large number of hail storms observed by newly upgraded dual-polarization WSR-88D systems, with ground truth collected as part of the NSSL SHAVE during 2012 and 2013 (Ortega et al. 2009, 2012). Following the SHAVE method, a team of meteorology students from the Uni- versity of
An example of the HSDA product with three cate- gories of hail is presented in Fig. 2. A severe hailstorm producing giant hail with diameters exceeding 10 cm was observed with the KFWS WSR-88D west of
The methods for discrimination of hail size at C and X bands can be devised on similar principles. The challenges are that 1) attenuation/differential attenu- ation is stronger at shorter wavelengths, 2) Z does not increase that much with maximal hail size as it does at S band and all radar variables are significantly less sen- sitive to hail with diameters larger than 25 mm (see Figs. 8a and 12 in Part I), and 3) ZDR may be equally high for smaller and larger hail. Some additional dis- crimination parameters can be utilized, however. For example, Kaltenboeck and Ryzhkov (2013) have found that the spatial variability of ZDR and rhv within high- reflectivity cores at C band may serve as additional useful parameters for discrimination between different hail sizes. The variability of ZDR and rhv increases with increasing hail size because of larger differential attenuation and lower rhv.
3. Polarimetric attenuation correction in melting hail
Melting hail produces strong attenuation/differential attenuation at S, C, and X bands. The relation between specific attenuation Ah and radar reflectivity in hail is different from the one in rain. We use the results of modeling in Part I to obtain best-fit power-law Ah(Z) relations in melting hail at S, C, and X bands and to compare them with the corresponding relations for pure rain retrieved from simulations using 47 144 drop size distributions measured in central
The polarimetric methods for attenuation correction are based on the use of the relations (Bringi et al. 1990)
where a 5 Ah/KDP and b 5 A DP/ K DP. The factors a and b in rain and their variability are well known. They are functions of ZDR and raindrop temperature. These de- pendences are illustrated in Fig. 4, where the scatter- plots of a and b versus ZDR for temperatures 08 and 308C simulated from the
The values of Ah, ADP, a, and b are generally higher in melting hail than in rain. Attenuation/differential attenuation in melting hail can be significant even at S band. An example of anomalously large attenuation/ differential attenuation experienced by the S-band KICT WSR-88D in a supercell storm east of
Utilizing Eq. (1) and the values of a and b accepted for convective rain at S band is not sufficient to com- pletely eliminate the biases in Z and ZDR induced by attenuation because the factors a and b are much higher in hail. Herein, we suggest a new method for attenuation correction in hail that is based on the ideas first presented by Carey et al. (2000), Ryzhkov et al. (2007, 2014), and Gu et al. (2011). According to this method, the hail-bearing convective core should be treated as a hot spot, and attenuation correction has to be performed separately using different factors a and b in hail and surrounding rain. It is assumed that a and b in rain are equal to their average climatological values, whereas a and b inhailmayvaryfromraytorayand should be determined from the radar data behind the hail cell.
As a starting point, a propagation path through pre- cipitation (r0, rm) containing hail in the range inter- val (r1, r2) has to be segmented into three parts: (r0, r1), (r1, r2), and (r2, rm). We use the Z threshold of 50 dBZ to identify the segment (r1, r2). Within a hot spot or hail segment (r1, r2), specific attenuation Ah is determined using the so-called ZPHI method (Testud et al. 2000; Bringi and Chandrasekar 2001):
... where (2)
the parameter b is the exponent in the Ah(Z) relation
valid in hail, and PIA is the two-way path-integrated attenuation within hail
In Eqs. (2)-(4), Za is the attenuated (biased) radar re- flectivity factor.
To find PIA in hail, we have to estimate the difference between a ''true'' (not biased by attenuation) value of Z(r2) and its measured value Za(r2). This can be done using the ZPHI estimate [Eq. (2)] of Ah within rain in- terval (r2, rm), where
As shown by Ryzhkov et al. (2014), the estimate of Ah(r2) is not affected by Z bias caused by radar mis- calibration, attenuation, partial beam blockage, or wet radome. Hence, the unbiased value of Z(r2) can be es- timated from Ah(r2) using Eq. (6). The intercept a in the Ah(Z) relation is notoriously prone to DSD variability and temperature. It is, however, reasonable to assume that the Ah(Z) relation does not vary much in rain sur- rounding a hail core; hence, the same Ah(Z) relation is valid for radials affected and unaffected by the presence of hail. In other words, intrinsic values of Z for a given Ah should be the same in the radials free of hail and the ones in the shadow of hail core. A similar assumption is utilized by Zhang et al. (2013) to correct reflectivity biases caused by partial beam blockage. After specific attenuation Ah is obtained in all three segments of the propagation path through precipitation (r0, rm), the corrected radar reflectivity factor can be computed as
After correction for attenuation, the area of Z . 50 dBZ appears extended farther away from the radar in the rear side of the hail cell and a hail-containing range interval expands so that its end range increases from r2 to r2(c). Then the previously described procedure should be re- peated after r2 is replaced by rm(c2) . In the radials that are free of hail, attenuation correction is performed using Eq. (1).
To correct differential reflectivity along the radials containing hail, the estimate of ZDR bias in the interval (r2, rm) should be made first. This is done by comparing the measured and expected (true) ZDR in the range gates where corrected Z is between 20 and 30 dBZ. The expected or reference ZDR in this range of reflectivities is obtained from the corrected Z as
The reason for using low reflectivities for such a com- parison is that the ZDR(Z) relation is less affected by DSD variability in lighter rain. Equation (10) has been obtained via simulations using the large DSD dataset collected in
Again, Eq. (1) can be used to correct ZDR along the radials that are free of hail.
The fields of Z and ZDR corrected for attenuation are shown in Figs. 5e and 5f. It is evident that the correction procedure efficiently eliminates the Z and ZDR biases caused by attenuation/differential attenuation. If total attenuation along the range interval (r0, r1) is negligible, then a path-integrated attenuation in hail PIA[r1, r2(c)]is equal to DZ where DZ 5 Z[r2(c)] 2 Za[r2(c)]. The m2agni- tude of DZ as a function of azimuth is displayed in Fig. 7 (thicker solid line) together with DFDP 5FDP[r(2c)] 2 FDP(r1) (thin solid line) and the bias in differential re- flectivity DZDR (dashed line). Figure 7 shows that the maximal absolute values of attenuation-induced biases of Z and ZDR are about 17 and 6 dB, respectively. Such anomalously high values of DZ and DZDR at S band correspond to relatively modest differential phase shift FDP barely exceeding 1508 (cf. Fig. 6c). The estimates of the factors a and b in hail can be obtained as the ratios ah 5DZ/DFDP and bh 5DZDR/DFDP.
Median values of ah and bh in the azimuthal sector from 878 to 928 are about 0.1 and 0.04 dB 821, respec- tively, which are an order of magnitude higher than typical values in rain (see Fig. 4). Theoretical simula- tions using the model 1 of melting hail described in Part I yield the values of ah and bh for large hail at S band within ranges (0.07-0.13 dB 821) and (0.013-0.027 dB 821) at the height levels 1.5-2 km below the freezing level. The maximal value of ah estimated from the radar measurements illustrated in Fig. 7 is just in the middle of the theoretical range for maximal hail size of 3.5 cm, which gives credibility to our theoretical model 1. The maximal observed bh is about 2 times that predicted by model 1 for this size of hail, however. This is not surprising because model 1 does not explicitly treat the effects of vigorous size sorting in rain that affect ZDR much more than Z. Additional size sorting, which is more adequately treated by the more sophisticated model 2, usually increases ZDR and the intensity of dif- ferential attenuation (i.e., magnitude of b; see Fig. 17 in Part I).
Radial profiles of Z and ZDR before and after cor- rection for attenuation along the radial with maximal attenuation at azimuth 5 88.78 are shown in Fig. 8. Corrected values of ZDR (bottom panel in Fig. 8) are lowest where Z is maximal within the hail core, which makes perfect sense (Bringi and Chandrasekar 2001). The minimal value of ZDR is about 0 dB where the corrected value of Z is 75 dBZ at a range of 28.5 km from the radar. Specific attenuation Ah in hail can be esti- mated by comparison of uncorrected and corrected ra- dial profiles of radar reflectivity. The Z bias of 17 dB is accumulated within a distance interval between r 5 18 and 30 km that corresponds to a net value of Ah of ;0.71 dB km21, which is more than an order of magni- tude higher than the maximal value in rain at S band. Notable are very high values of ZDR approaching 5-6 dB at the periphery of the hail cell, which are indicative of strong size sorting (Kumjian and Ryzhkov 2008; Ryzhkov et al. 2011; Kaltenboeck and Ryzhkov 2013).
Anomalously high attenuation is relatively rare at S band (even in hail). It is more common at C and X bands. Borowska et al. (2011) evaluated specific attenuation in melting hail at C band through direct comparisons with simultaneous measurements at S band and showed that Ah in wet hail is highly variable and can be an order of magnitude higher than in pure rain with an intensity of 100 mm h21. This conclusion is very similar to our findings at S band. Substantial differential attenuation at C band (up to 2 dB km21) was also reported in that study.
Vertical profiles of the factors a and b at C band simulated from model 1 are displayed in Figs. 9a and 9b in the cases of no hail (NH), small hail (SH), moderate hail (MH), and large hail (LH) (see definitions of these categories in Part I). Both factors exhibit a strong de- pendence on height, with maximal simulated values lo- cated at about 1.5-2.0 km below the freezing level. The specific attenuation Ah increases dramatically with in- creasing hail size, whereas ADP and KDP are less affected by hail and are primarily determined by large raindrops and small water-coated hailstones. Hence, the ratio a is more sensitive to the presence of hail than is b.
A comparison of the variability of ranges of a and b in pure rain and melting hail mixed with rain at C band is illustrated in Figs. 9c and 9d. It is assumed that the factor a varies between 0.05 and 0.18 dB 82 1 in pure rain, with a median value of 0.08 dB 821, whereas the factor b changes within the interval 0.008-0.1 dB 821,withamedian value 0.02 dB 821 (Ryzhkov et al. 2007; Tabary et al. 2009). The ranges of variability of simulated a and b for NH, SH, MH, and LH are shown as shaded polygons using the data displayed in Figs. 9a and 9b. It is apparent that the bulk of variability of a is due to the presence of hail while the factor b is already very variable in rain and the presence of hail does not add much to its overall variation. The simulated values of a and b are in good agreement with the estimates from observations by Ryzhkov et al. (2007) using the hot-spot method for at- tenuation correction that is described in Gu et al. (2011). The estimation of a and b from direct comparison of S- and C-band measurements with closely located radars by Borowska et al. (2011) shows generally higher values of the ratios a and b than are predicted by model 1.
Attenuation/differential attenuation correction in hail at C and X bands can be performed using the approach that is applied for S band in this study. Testing such an approach at shorter wavelengths is a subject for future investigation. The proposed method for attenuation correction in hail implies the estimation of the average magnitude of specific attenuation Ah within the hail core, which is sensitive to maximal hail size and can be potentially utilized in HSDA. This hypothesis requires further exploration.
4. Polarimetric rainfall estimation in the presence of hail
The observational data provide ample evidence that the rainfall estimation algorithm that is based on the use of specific differential phase KDP is the best choice in the situations of rain mixed with hail because KDP is relatively insensitive to the presence of hail (e.g., Ryzhkov et al. 2005). The results of our modeling study generally support this notion, although with certain reservations. Vertical profiles of true rain rates retrieved from the model of melting hail described in Part I and their R(KDP) estimates for different hail sizes at the three radar wavelengths are shown in Fig. 10. The R(KDP) relation has a power-law form:
where the factors a and b depend on radar wavelength. Theoretical simulations of R and KDP for a large set of DSD measurements in
We evaluated the performance of the Z- and KDP- based algorithms at S band for rainfall estimation in the mixture of rain and hail for the hailstorm on
where Z is capped at the 53-dBZ threshold. The po- larimetric algorithm combines the R(KDP) relation in Eq. (12) for heavy rain and hail with Z exceeding 45 dBZ and the R(Z) relation in Eq. (13) for Z less than 45 dBZ. The maps of 6-h rain totals retrieved by the two algo- rithms are displayed in Fig. 11. The conventional algo- rithm yields visibly higher overall rain totals than the polarimetric algorithm and tends to overestimate the actual rain accumulation reported by gauges, as Fig. 12 shows. This overestimation is particularly significant for higher rain totals. In contrast, the polarimetric algorithm utilizing KDP in hail cores produces almost unbiased estimates of rain accumulation over a whole range of rain totals (right panel in Fig. 12). The bias of the 6-h rain total estimate has been reduced from 27% down to 4%, whereas the fractional RMS error decreases from 32% to 19% if the combination of R(KDP) and R(Z)is utilized instead of the stand-alone R(Z) relation.
Note that in the middle of cells containing large hail exceeding 25 mm in size, the R(KDP) relation may overestimate rain as predicted by the simulations shown in Fig. 10. Indeed, measured KDP values that are over 78km21 than 218 mm h21, which is apparently too high, even for the case of severe hail. Such local overestimation does not have much of an impact on the overall performance of the KDP-based algorithm for estimation of hourly or multihour rainfall accumulations (Fig. 12), how- ever. Anomalously high local values of KDP exceeding 68-78 km21 at S band may be used as indication of large or giant hail to complement the algorithm for hail detection and determination of its size described in Part I.
The advantage of utilizing KDP for quantification of rain in mixture with hail was also demonstrated at X band in the recent study by Matrosov et al. (2013), where the relation R(KDP) 5 14KD0:7P7 was used. It has to be kept in mind that the KDP data in hail can be very noisy if the cross-correlation coefficient rhv drops too low, which is more likely at shorter wavelengths. Backscatter differ- ential phase d in melting hail can be significant, ranging from 58 to 158 according to our simulations (not shown). Hence, large positive and negative excursions of the KDP estimate may occur if the effects of forward prop- agation and backscattering are not separated in the measurements of differential phase. Such problems with accurate estimation of KDP are particularly pronounced at C band, where large raindrops resulting from melted hailstones cause appreciable resonance scattering effects.
Any polarimetric algorithm for detection of hail and determination of its size has to take into account the height of the center of the radar resolution volume with respect to the freezing level. The algorithm should be also radar wavelength-dependent because the radar variables at C and X bands are much less sensitive to the presence of large hail with sizes exceeding 2.5 cm than at S band, and the effects of resonance scattering differ- ently impact the radar variables at the three frequency bands. A new hail size discrimination algorithm is sug- gested and implemented at S band. The algorithm uti- lizes Z, ZDR, and rhv and is designed to distinguish between small hail (with diameter D less than 2.5 cm), large hail (2.5 , D , 5.0 cm), and giant hail with size exceeding 5.0 cm. The algorithm is based on the princi- ples of fuzzy logic. Different membership functions for each of the three radar variables are utilized in six height intervals, two of which are above the freezing level and four of them are below the freezing level. The results of validation tests for the newly proposed algorithm will be presented in Part III of this study.
Melting hail may significantly enhance attenuation/ differential attenuation of the radar signal, even at S band. A new method for attenuation correction is in- troduced and tested for the case of severe hail observed by the dual-polarization S-band WSR-88D. The exam- ined hailstorm produced biases of Z and ZDR up to 17 and 6 dB, respectively. The method allows one to quantify the attenuation-correction factors a 5 Ah/KDP and b 5 ADP/KDP in hail, which turned out to be an order of magnitude higher than typical values in rain at S band. The values of Ah and a are in good agreement with results of theoretical simulations using the model 1 described in Part I. The observed maximal values of b and ZDR are higher than predicted from model 1 and are in better agreement with predictions from the more sophisticated model 2, which takes into account the ef- fects of size sorting and particle collisions. Theoretical values of a and b in hail at C band simulated by model 1 are generally consistent with but somewhat lower than their estimates in the experimental studies of Ryzhkov et al. (2007) and Borowska et al. (2011).
It is shown that the R(KDP) relation is a good choice for estimating rainfall in the presence of hail at all three microwave frequency bands, but it may produce positive bias at S band for larger-sized hail. Such local over- estimation of rain mixed with large hail may not com- promise the accuracy of hourly or multihour rain totals obtained for hail-bearing storms, however.
Acknowledgments. Funding for the study was pro- vided by NOAA/
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ALEXANDER V. RYZHKOV,MATTHEW R. KUMJIAN,*
* Current affiliation: Advanced Study Program,
Corresponding author address: Dr.
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